L10a120

L10a119

L10a121

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L10a120 at Knotilus!

Link Presentations

[edit Notes on L10a120's Link Presentations]

Planar diagram presentation	X12,1,13,2 X2,13,3,14 X14,3,15,4 X16,5,17,6 X6,11,7,12 X20,7,11,8 X18,9,19,10 X10,17,1,18 X8,19,9,20 X4,15,5,16
Gauss code	{1, -2, 3, -10, 4, -5, 6, -9, 7, -8}, {5, -1, 2, -3, 10, -4, 8, -7, 9, -6}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	${\frac {u^{4}\left(-v^{2}\right)-u^{3}v^{3}+u^{3}v^{2}-u^{3}v-u^{2}v^{4}+u^{2}v^{3}-u^{2}v^{2}+u^{2}v-u^{2}-uv^{3}+uv^{2}-uv-v^{2}}{u^{2}v^{2}}}$ (db)
Jones polynomial	$-{\frac {2}{q^{9/2}}}+{\frac {1}{q^{7/2}}}-{\frac {1}{q^{5/2}}}-{\frac {1}{q^{25/2}}}+{\frac {1}{q^{23/2}}}-{\frac {2}{q^{21/2}}}+{\frac {3}{q^{19/2}}}-{\frac {4}{q^{17/2}}}+{\frac {4}{q^{15/2}}}-{\frac {4}{q^{13/2}}}+{\frac {3}{q^{11/2}}}$ (db)
Signature	-5 (db)
HOMFLY-PT polynomial	$a^{11}z^{3}+3a^{11}z+a^{11}z^{-1}-a^{9}z^{5}-4a^{9}z^{3}-4a^{9}z-a^{9}z^{-1}-a^{7}z^{5}-3a^{7}z^{3}-a^{7}z-a^{5}z^{5}-4a^{5}z^{3}-3a^{5}z$ (db)
Kauffman polynomial	$a^{15}z^{5}-4a^{15}z^{3}+3a^{15}z+a^{14}z^{6}-3a^{14}z^{4}+a^{14}z^{2}+a^{13}z^{7}-3a^{13}z^{5}+2a^{13}z^{3}-a^{13}z+a^{12}z^{8}-4a^{12}z^{6}+6a^{12}z^{4}-3a^{12}z^{2}+a^{11}z^{9}-6a^{11}z^{7}+15a^{11}z^{5}-15a^{11}z^{3}+7a^{11}z-a^{11}z^{-1}+2a^{10}z^{8}-10a^{10}z^{6}+18a^{10}z^{4}-8a^{10}z^{2}+a^{10}+a^{9}z^{9}-6a^{9}z^{7}+15a^{9}z^{5}-15a^{9}z^{3}+7a^{9}z-a^{9}z^{-1}+a^{8}z^{8}-4a^{8}z^{6}+6a^{8}z^{4}-3a^{8}z^{2}+a^{7}z^{7}-3a^{7}z^{5}+2a^{7}z^{3}-a^{7}z+a^{6}z^{6}-3a^{6}z^{4}+a^{6}z^{2}+a^{5}z^{5}-4a^{5}z^{3}+3a^{5}z$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 χ -4                     1 1 -6                   1 1 0 -8                 1     1 -10               2 1     -1 -12             2 1       1 -14           2 2         0 -16         2 2           0 -18       1 2             1 -20     1 2               -1 -22     1                 1 -24 1 1                   0 -26 1                     1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-6$ $i=-4$ $r=-10$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-9$ ${\mathbb {Z} }$ $r=-8$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-6$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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